Adaptive quadrature

In topology, a preclosure operator, or Čech closure operator is a map between subsets of a set, similar to a topological closure operator, except that it is not required to be idempotent. That is, a preclosure operator obeys only three of the four Kuratowski closure axioms.

Definition

A preclosure operator on a set $X$ is a map $[{]}_p$

$[{]}_p:\mathcal{𝒫}(X)\to \mathcal{𝒫}(X)$

where $\mathcal{𝒫}(X)$ is the power set of $X$.

The preclosure operator has to satisfy the following properties:

1. $[\varnothing {]}_p=\varnothing$ (Preservation of nullary unions);
2. $A\subseteq [A{]}_p$ (Extensivity);
3. $[A\cup B{]}_p=[A{]}_p\cup [B{]}_p$ (Preservation of binary unions).

The last axiom implies the following:

4. $A\subseteq B$ implies $[A{]}_p\subseteq [B{]}_p$.

Topology

A set $A$ is closed (with respect to the preclosure) if $[A{]}_p=A$. A set $U\subset X$ is open (with respect to the preclosure) if $A=X\setminus U$ is closed. The collection of all open sets generated by the preclosure operator is a topology.

The closure operator cl on this topological space satisfies $[A{]}_p\subseteq \mathrm{cl}(A)$ for all $A\subset X$.

Examples

Premetrics

Given $d$ a premetric on $X$, then

$[A{]}_p=\{x\in X:d(x,A)=0\}$

is a preclosure on $X$.

Sequential spaces

The sequential closure operator $[{]}_{\text{seq}}$ is a preclosure operator. Given a topology $\mathcal{𝒯}$ with respect to which the sequential closure operator is defined, the topological space $(X,\mathcal{𝒯})$ is a sequential space if and only if the topology ${\mathcal{𝒯}}_{\text{seq}}$ generated by $[{]}_{\text{seq}}$ is equal to $\mathcal{𝒯}$, that is, if ${\mathcal{𝒯}}_{\text{seq}}=\mathcal{𝒯}$.

See also

• Eduard Čech

References

• A.V. Arkhangelskii, L.S.Pontryagin, General Topology I, (1990) Springer-Verlag, Berlin. ISBN 3-540-18178-4.
• B. Banascheski, Bourbaki's Fixpoint Lemma reconsidered, Comment. Math. Univ. Carolinae 33 (1992), 303-309.